The standard form of the equation of a hyperbola with
center (h, k) and transverse axis parallel to the y-axis is
[tex]\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1[/tex]
The coordinates of the foci are
[tex](h,k\pm c)[/tex]
Where c is
[tex]c^2=a^2+b^2[/tex]
Since the given equation is
[tex]\frac{(y-1)^2}{9}-\frac{(x-3)^2}{16}=1[/tex]
Compare it with the form above, then
[tex]\begin{gathered} h=3 \\ k=1 \\ a^2=9 \\ b^2=16 \end{gathered}[/tex]
Let us find c by using the rule above
[tex]\begin{gathered} c^2=9+16 \\ c^2=25 \\ c=\pm\sqrt[]{25} \\ c=\pm5 \end{gathered}[/tex]
Substitute the values of h, k, c in the coordinates of the foci above
[tex]\begin{gathered} (3,1+5),(3,1-5) \\ (3,6),(3,-4) \end{gathered}[/tex]
The coordinates of the foci are (3,6) and (3,-4)
